Introduction

Welcome

Aims for this deep dive:

• Share some challenges from mechanical engineering
• Provide a benchmark problem for you to work on
• Promote some interdisciplinary collaboration

Outline For The Day

Today:

• Introduction to the problem setting and challenges
• Tutorial on accessing the data
• Demonstration of some basic approaches:
  • System ID
  • Damage detection
• Examples of some more advanced things we could do
• Discussion of what you could get involved with

Outline For The Day

Rest of the week

Our Benchmark Dataset

A Bit of Background Theory

A Bit of Background Theory

\[ \newcommand{\identity}{\mathbb{I}} \renewcommand{\vec}[1]{\boldsymbol{#1}} \newcommand{\mat}[1]{{#1}} \newcommand{\transpose}{^\text{T}} \newcommand{\inv}{^{-1}} \]

\[ M\boldsymbol{\ddot{x}}+C\boldsymbol{\dot{x}}+K\boldsymbol{x} = \boldsymbol{F}(t) \]

Free Vibration Response

Taking the undamped system equation in free vibration, \[ M\boldsymbol{\ddot{x}}+K\boldsymbol{x} = 0 \]

Assuming a harmonic solution \(x_i = X_i \sin (\omega t)\), we arrive at an eigenvalue problem.

\[ \underbrace{M^{-1}K}_A \boldsymbol{X} = \lambda \boldsymbol{X} \]

Forced Vibration Response

Forced Vibration Response

Remember for an SDOF system, the FRF is defined as: \[ H(\nu) = \frac{X(\nu)}{F(\nu)} \]

Forced Vibration Response

Forced Vibration Response

Modal Decomposition

Remember we have, \[\begin{equation*} \mat{M}\vec{\ddot{x}} + \mat{C}\vec{\dot{c}} + \mat{K}\vec{x} = \vec{F} \end{equation*}\]

Substituting \(\vec{x} = \mat{\tilde{\Phi}}\vec{y}\) and premultiplying by \(\mat{\tilde{\Phi}}\transpose\), \[\begin{equation*} \mat{\tilde{\Phi}}\transpose \mat{M} \mat{\tilde{\Phi}}\vec{\ddot{y}} + \mat{\tilde{\Phi}}\transpose \mat{C} \mat{\tilde{\Phi}}\vec{\dot{y}} + \mat{\tilde{\Phi}}\transpose \mat{K} \mat{\tilde{\Phi}}\vec{y} = \mat{\tilde{\Phi}}\transpose\vec{F} \end{equation*}\]

Which gives,

\[\begin{equation*} \identity\vec{\ddot{y}} + \mat{2\zeta\Lambda^{1/2}}\vec{\dot{y}} +\mat{\Lambda}\vec{y} = \mat{\tilde{\Phi}}\transpose\vec{F} \end{equation*}\]

Analysing Large Scale Systems

\[\begin{equation*} H_{jk}(\nu) = \sum_{r=1}^R\frac{\varphi_j^{(r)}\varphi_k^{(r)}}{\mat{K_{rr}} - \nu^2\mat{M_{rr}}+ i \nu\mat{C_{rr}}} \end{equation*}\]

Quick Recap On The Aims

Components of Interest

To make this a bit more concrete we need:

• A model of the system we can trust
• The ability to infer important information about the system, e.g. is it damaged/not performing as expected?
• To account for time varying effects in the system
• Methods for making effective decisions based on this information

Tasks for this Deep Dive

Identification I

The core task here is to establish a model which can:

1. Replicate the dynamic behaviour of the system mathematically
2. Provide insight into the physical mechanisms behind the response
3. Quantify our uncertainty in the system (perhaps consider if that uncertainty is epistemic or alleatoric)

Identification II

To approach this the common methods would:

• Assume linearity
• Assume “modal behaviour” - i.e. the system can be well described by a subset of its eigenmodes
• Look for deterministic solution
• Assume stationarity
• Consider a discretised version of the system, i.e. ODEs not PDEs
• Adopt a low order model

Identification III

Challenges to try should focus on providing one of (a)-(c) as listed above, for example:

• Developing a low order model of the system as a set of ODEs/PDEs
• Validation of established models for variable conditions applied in testing
• Determining if the system is nonlinear, if so, providing physical models of that nonlinearity supported by data

Prediction

Why do we care about prediction:

• Replicate unseen behaviour of interest
• Explore extreme events or scenarios
• Infer unmeasurable data in operation (virtual sensing)
• Validation of identified models
• Allow for comparison between models

Prediction: Virtual Sensing

In summary we can think of the problem as shown below:

Prediction: Load Estimation

The issue with ambient loads:

1. Cannot be controlled for the purpose of system identification
2. Cannot be feasibly measured
3. Are the major root cause of damage to the structure

Physics-Informed Machine Learning

Problems with using ML:

• Difficulty in extrapolation
• Noisy measurements and hidden states
• Model selection and complexity
• Validation of models

Evaluation

In the evaluation task we would like to understand the condition of the structure. Remembering Rytter’s hierarchy:

In this task we will look at methods that span this set of challenges.

Detection

Difficulties:

• Varying test conditions can appear as damage
• We assume have never seen damage until it happens
• The effect from damage may be much less but similar to other factors, e.g. nonlinearity

Localisation

Difficulties:

• If the effect of damage is unknown this can be hard to localise
• Damage will almost always be appearing in a new location we don’t have examples from
• We would like to localise the smallest possible amount of damage to ensure early action can be taken

A Very Valuable Output

Parameters of this model may include:

• Damage location
• Damage extent
• Different sensing systems
• Longer term forecasts

Loading Data

We provide a repo of useful tools under hawk_tools

data = hawk_tools.get_hawk_data(
    test_camp='LMS', 
    test_id='BR_AR', 
    test_runs=1, 
    test_reps=[1,2], 
    download_dir="../.hawk_data", 
    disk_only=False, 
    ask=True, 
    load_kwargs={'compress_x_axis': True}
)

Loading Options I

Choice of test series:

Loading Options II

Loading Options III

Demo

Load a single repetition of a single test

data = hawk_tools.get_hawk_data("LMS", "BR_AR", 1, 1)
print(list(data.keys()))  # available datasets
print(list(data["BR_AR_1_1"].keys()))  # available sensors in the data
print(list(data["BR_AR_1_1"]["LTC-03"].keys()))  # fields per sensor

Using cached version: ./.hawk_data\LMS_BR_AR_1_1.hd5

['BR_AR_1_1']
['EXH', 'FRC', 'LLC-01', 'LLC-02', 'LLC-03', 'LLC-04', 'LLC-05', 'LLC-06', 'LLC-07', 'LLG-01', 'LLG-02', 'LLG-03', 'LLG-04', 'LLG-05', 'LLG-06', 'LRT-01', 'LRT-02', 'LRT-03', 'LRT-04', 'LTC-01', 'LTC-02', 'LTC-03', 'LTC-04', 'LTC-05', 'LTC-06', 'LTC-07', 'Meta', 'TRI-X', 'TRI-Y', 'TRI-Z', 'ULC-01', 'ULC-02', 'ULC-03', 'ULC-04', 'ULC-05', 'ULC-06', 'ULC-07', 'ULE-01', 'ULE-02', 'ULE-03', 'ULE-04', 'ULE-05', 'ULE-06', 'ULE-07', 'URT-01', 'URT-02', 'URT-03', 'UTC-01', 'UTC-02', 'UTC-03', 'UTC-04', 'UTC-05', 'UTC-06', 'UTC-07', 'UTE-01', 'UTE-02', 'UTE-03', 'UTE-04', 'UTE-05', 'UTE-06', 'UTE-07']
['Autopower Spectrum', 'Coherence Spectrum', 'Dynamic Stiffness', 'Frequency Response Function', 'Spectra', 'Time Signal', 'assettNumber', 'calibration', 'location', 'model', 'sensor']

Plotting an FRF

H = data["BR_AR_1_1"]["LTC-03"]["Frequency Response Function"]["Y_data"]["value"]
w = data["BR_AR_1_1"]["LTC-03"]["Frequency Response Function"]["X_data"]["value"]

H_units = data["BR_AR_1_1"]["LTC-03"]["Frequency Response Function"]["Y_data"]["units"]

fig, axs = plt.subplots(2, 1, figsize=(8, 5))
axs[0].semilogy(w, np.abs(H))
axs[1].plot(w, np.angle(H))
axs[0].set_xlabel(r"$\omega$ (Hz)")
axs[1].set_xlabel(r"$\omega$ (Hz)")
axs[0].set_ylabel(r"$|H(\omega)|$ ({})".format(H_units))
axs[1].set_ylabel(r"$\angle H(\omega)$ ({})".format(H_units))
axs[0].set_xlim([0, 160])
axs[1].set_xlim([0, 160])
plt.tight_layout()
plt.show()

Some More Data Examples

outdir = "../.hawk_data"

# %% load more repetitions - note that the earlier download is cached
data = hawk_tools.get_hawk_data(
    "LMS",
    "BR_AR",
    1,
    [1, 2, 3],
    download_dir=outdir)
print(list(data.keys()))

Using cached version: ../.hawk_data\LMS_BR_AR_1_2.hd5

Using cached version: ../.hawk_data\LMS_BR_AR_1_1.hd5

Using cached version: ../.hawk_data\LMS_BR_AR_1_3.hd5

['BR_AR_1_2', 'BR_AR_1_1', 'BR_AR_1_3']

Now let’s look at the NI test campaign

# %% Load some NI data
data = hawk_tools.get_hawk_data("NI", "RPH_AR", 1, [1, 2], download_dir=outdir)
print(list(data.keys()))  # available datasets
print(list(data["RPH_AR_1_1"].keys()))  # available sensors in the data
print(list(data["RPH_AR_1_1"]["LTC-01"].keys()))  # fields per sensor

Using cached version: ../.hawk_data\NI_RPH_AR_1_1.hd5
Using cached version: ../.hawk_data\NI_RPH_AR_1_2.hd5

['RPH_AR_1_1', 'RPH_AR_1_2']
['EXH', 'FRC-01', 'LLC-01', 'LLC-02', 'LLC-03', 'LLC-04', 'LLC-05', 'LLC-06', 'LLC-07', 'LLG-01', 'LLG-02', 'LLG-03', 'LLG-04', 'LLG-05', 'LLG-06', 'LRT-01', 'LRT-02', 'LRT-03', 'LRT-04', 'LTC-01', 'LTC-02', 'LTC-03', 'LTC-04', 'LTC-05', 'LTC-06', 'LTC-07', 'TRI-X', 'TRI-Y', 'TRI-Z', 'ULC-01', 'ULC-02', 'ULC-03', 'ULC-04', 'ULC-05', 'ULC-06', 'ULC-07', 'ULE-01', 'ULE-02', 'ULE-03', 'ULE-04', 'ULE-05', 'ULE-06', 'ULE-07', 'URT-01', 'URT-02', 'URT-03', 'UTC-01', 'UTC-02', 'UTC-03', 'UTC-04', 'UTC-05', 'UTC-06', 'UTC-07', 'UTE-01', 'UTE-02', 'UTE-03', 'UTE-04', 'UTE-05', 'UTE-06', 'UTE-07', 'Meta']
['Measurement', 'Timestamp']

Advanced Usage

load_opts = {
    "data": "Frequency Response Function",  # FRFs only
    "meta": False,  # do not load metadata i.e test specific data
    "attrs": False,  # do not load attributes i.e sensor number
    "compress_x_axis": True,  # add x data as seperate channel (rather than copying to each sensor)
    "sensors": None,  # Load all sensors
}

data = hawk_tools.get_hawk_data(
    "LMS",
    "BR_AR",
    1,
    [1, 2, 3],
    load_kwargs=load_opts,
    download_dir=outdir)

print(list(data.keys()))  # available datasets
print(list(data["BR_AR_1_1"].keys()))  # available sensors in the data
print(list(data["BR_AR_1_1"]["LTC-03"].keys()))  # fields per sensor

ws = data["BR_AR_1_1"][
    "X_data"
]  # frequency vector available at the top level of each test now

Using cached version: ../.hawk_data\LMS_BR_AR_1_2.hd5
Using cached version: ../.hawk_data\LMS_BR_AR_1_1.hd5
Using cached version: ../.hawk_data\LMS_BR_AR_1_3.hd5

['BR_AR_1_2', 'BR_AR_1_1', 'BR_AR_1_3']
['EXH', 'X_data', 'FRC', 'LLC-01', 'LLC-02', 'LLC-03', 'LLC-04', 'LLC-05', 'LLC-06', 'LLC-07', 'LLG-01', 'LLG-02', 'LLG-03', 'LLG-04', 'LLG-05', 'LLG-06', 'LRT-01', 'LRT-02', 'LRT-03', 'LRT-04', 'LTC-01', 'LTC-02', 'LTC-03', 'LTC-04', 'LTC-05', 'LTC-06', 'LTC-07', 'Meta', 'TRI-X', 'TRI-Y', 'TRI-Z', 'ULC-01', 'ULC-02', 'ULC-03', 'ULC-04', 'ULC-05', 'ULC-06', 'ULC-07', 'ULE-01', 'ULE-02', 'ULE-03', 'ULE-04', 'ULE-05', 'ULE-06', 'ULE-07', 'URT-01', 'URT-02', 'URT-03', 'UTC-01', 'UTC-02', 'UTC-03', 'UTC-04', 'UTC-05', 'UTC-06', 'UTC-07', 'UTE-01', 'UTE-02', 'UTE-03', 'UTE-04', 'UTE-05', 'UTE-06', 'UTE-07']
['Frequency Response Function']

A Quick Look at The Data

data = hawk_tools.get_hawk_data(
    "LMS",
    "BR_AR",
    1,
    1,
    load_kwargs=load_opts,
    download_dir=outdir)

def plot_bode_frf(axs, w, H):
    axs[0].semilogy(w, np.abs(H), linewidth=0.2)
    axs[1].plot(w, np.angle(H), linewidth=0.2)


fig, axs = plt.subplots(2, 1, figsize=(8, 5))
for k, v in data['BR_AR_1_1'].items():
    if 'U' in k:
        plot_bode_frf(
            axs,
            data['BR_AR_1_1']['X_data']['Frequency Response Function']['value'],
            v['Frequency Response Function']['Y_data']['value'])

axs[0].set_xlabel(r"$\omega$ (Hz)")
axs[1].set_xlabel(r"$\omega$ (Hz)")
axs[0].set_ylabel(r"$|H(\omega)|$")
axs[1].set_ylabel(r"$\angle H(\omega)$")
axs[0].set_xlim([0, 160])
axs[1].set_xlim([0, 160])

plt.tight_layout()
plt.show()

Using cached version: ../.hawk_data\LMS_BR_AR_1_1.hd5

Getting Setup

Broadly:

• Setup a venv!
• git clone https://github.com/MDCHAMP/INI_hawk_notebooks
• Run pip install -r INI_hawk_requirements.txt
• Open notebook notebooks/01_Hawk_intro.ipynb

What’s Next

• Some theory on analysing dynamic data
• How to extract some useful information from this messy data
• Can that information be used to determine damage?
• Coffee!